unit real number system homework 4 rational vs irrational numbers

1.1 Real Numbers: Algebra Essentials

Learning objectives.

In this section, you will:

• Classify a real number as a natural, whole, integer, rational, or irrational number.
• Perform calculations using order of operations.
• Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity.
• Evaluate algebraic expressions.
• Simplify algebraic expressions.

It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.

Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.

But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century CE in India that zero was added to the number system and used as a numeral in calculations.

Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century CE, negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.

Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.

Classifying a Real Number

The numbers we use for counting, or enumerating items, are the natural numbers : 1, 2, 3, 4, 5, and so on. We describe them in set notation as { 1 , 2 , 3 , ... } { 1 , 2 , 3 , ... } where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers . Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero: { 0 , 1 , 2 , 3 , ... } . { 0 , 1 , 2 , 3 , ... } .

The set of integers adds the opposites of the natural numbers to the set of whole numbers: { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.

The set of rational numbers is written as { m n | m and  n are integers and  n ≠ 0 } . { m n | m and  n are integers and  n ≠ 0 } . Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed as a terminating or repeating decimal. Any rational number can be represented as either:

• ⓐ a terminating decimal: 15 8 = 1.875 , 15 8 = 1.875 , or
• ⓑ a repeating decimal: 4 11 = 0.36363636 … = 0. 36 ¯ 4 11 = 0.36363636 … = 0. 36 ¯

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

Writing Integers as Rational Numbers

Write each of the following as a rational number.

Write a fraction with the integer in the numerator and 1 in the denominator.

• ⓐ 7 = 7 1 7 = 7 1
• ⓑ 0 = 0 1 0 = 0 1
• ⓒ −8 = − 8 1 −8 = − 8 1

Identifying Rational Numbers

Write each of the following rational numbers as either a terminating or repeating decimal.

• ⓐ − 5 7 − 5 7
• ⓑ 15 5 15 5
• ⓒ 13 25 13 25

Write each fraction as a decimal by dividing the numerator by the denominator.

• ⓐ − 5 7 = −0. 714285 ——— , − 5 7 = −0. 714285 ——— , a repeating decimal
• ⓑ 15 5 = 3 15 5 = 3 (or 3.0), a terminating decimal
• ⓒ 13 25 = 0.52 , 13 25 = 0.52 , a terminating decimal
• ⓐ 68 17 68 17
• ⓑ 8 13 8 13
• ⓒ − 17 20 − 17 20

Irrational Numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even 3 2 , 3 2 , but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers . Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.

Differentiating Rational and Irrational Numbers

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

• ⓑ 33 9 33 9
• ⓓ 17 34 17 34
• ⓔ 0.3033033303333 … 0.3033033303333 …
• ⓐ 25 : 25 : This can be simplified as 25 = 5. 25 = 5. Therefore, 25 25 is rational.

So, 33 9 33 9 is rational and a repeating decimal.

• ⓒ 11 : 11 11 : 11 is irrational because 11 is not a perfect square and 11 11 cannot be expressed as a fraction.

So, 17 34 17 34 is rational and a terminating decimal.

• ⓔ 0.3033033303333 … 0.3033033303333 … is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.
• ⓐ 7 77 7 77
• ⓒ 4.27027002700027 … 4.27027002700027 …
• ⓓ 91 13 91 13

Real Numbers

Given any number n , we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers . As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.

The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line as shown in Figure 1 .

Classifying Real Numbers

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

• ⓐ − 10 3 − 10 3
• ⓒ − 289 − 289
• ⓓ −6 π −6 π
• ⓔ 0.615384615384 … 0.615384615384 …
• ⓐ − 10 3 − 10 3 is negative and rational. It lies to the left of 0 on the number line.
• ⓑ 5 5 is positive and irrational. It lies to the right of 0.
• ⓒ − 289 = − 17 2 = −17 − 289 = − 17 2 = −17 is negative and rational. It lies to the left of 0.
• ⓓ −6 π −6 π is negative and irrational. It lies to the left of 0.
• ⓔ 0.615384615384 … 0.615384615384 … is a repeating decimal so it is rational and positive. It lies to the right of 0.
• ⓑ −11.411411411 … −11.411411411 …
• ⓒ 47 19 47 19
• ⓓ − 5 2 − 5 2
• ⓔ 6.210735 6.210735

Sets of Numbers as Subsets

Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as Figure 2 .

Sets of Numbers

The set of natural numbers includes the numbers used for counting: { 1 , 2 , 3 , ... } . { 1 , 2 , 3 , ... } .

The set of whole numbers is the set of natural numbers plus zero: { 0 , 1 , 2 , 3 , ... } . { 0 , 1 , 2 , 3 , ... } .

The set of integers adds the negative natural numbers to the set of whole numbers: { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } .

The set of rational numbers includes fractions written as { m n | m and  n are integers and  n ≠ 0 } . { m n | m and  n are integers and  n ≠ 0 } .

The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: { h | h is not a rational number } . { h | h is not a rational number } .

Differentiating the Sets of Numbers

Classify each number as being a natural number ( N ), whole number ( W ), integer ( I ), rational number ( Q ), and/or irrational number ( Q′ ).

• ⓔ 3.2121121112 … 3.2121121112 …
• ⓐ − 35 7 − 35 7
• ⓔ 4.763763763 … 4.763763763 …

Performing Calculations Using the Order of Operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example, 4 2 = 4 ⋅ 4 = 16. 4 2 = 4 ⋅ 4 = 16. We can raise any number to any power. In general, the exponential notation a n a n means that the number or variable a a is used as a factor n n times.

In this notation, a n a n is read as the n th power of a , a , or a a to the n n where a a is called the base and n n is called the exponent . A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, 24 + 6 ⋅ 2 3 − 4 2 24 + 6 ⋅ 2 3 − 4 2 is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations . This is a sequence of rules for evaluating such expressions.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.

The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.

Let’s take a look at the expression provided.

There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify 4 2 4 2 as 16.

Next, perform multiplication or division, left to right.

Lastly, perform addition or subtraction, left to right.

Therefore, 24 + 6 ⋅ 2 3 − 4 2 = 12. 24 + 6 ⋅ 2 3 − 4 2 = 12.

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

Order of Operations

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS :

P (arentheses) E (xponents) M (ultiplication) and D (ivision) A (ddition) and S (ubtraction)

Given a mathematical expression, simplify it using the order of operations.

• Step 1. Simplify any expressions within grouping symbols.
• Step 2. Simplify any expressions containing exponents or radicals.
• Step 3. Perform any multiplication and division in order, from left to right.
• Step 4. Perform any addition and subtraction in order, from left to right.

Using the Order of Operations

Use the order of operations to evaluate each of the following expressions.

• ⓐ ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 )
• ⓑ 5 2 − 4 7 − 11 − 2 5 2 − 4 7 − 11 − 2
• ⓒ 6 − | 5 − 8 | + 3 ( 4 − 1 ) 6 − | 5 − 8 | + 3 ( 4 − 1 )
• ⓓ 14 − 3 ⋅ 2 2 ⋅ 5 − 3 2 14 − 3 ⋅ 2 2 ⋅ 5 − 3 2
• ⓔ 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1
• ⓐ ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) = ( 6 ) 2 − 4 ( 8 ) Simplify parentheses. = 36 − 4 ( 8 ) Simplify exponent. = 36 − 32 Simplify multiplication. = 4 Simplify subtraction. ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) = ( 6 ) 2 − 4 ( 8 ) Simplify parentheses. = 36 − 4 ( 8 ) Simplify exponent. = 36 − 32 Simplify multiplication. = 4 Simplify subtraction.

Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.

• ⓒ 6 − | 5 − 8 | + 3 | 4 − 1 | = 6 − | −3 | + 3 ( 3 ) Simplify inside grouping symbols. = 6 - ( 3 ) + 3 ( 3 ) Simplify absolute value. = 6 - 3 + 9 Simplify multiplication. = 12 Simplify addition. 6 − | 5 − 8 | + 3 | 4 − 1 | = 6 − | −3 | + 3 ( 3 ) Simplify inside grouping symbols. = 6 - ( 3 ) + 3 ( 3 ) Simplify absolute value. = 6 - 3 + 9 Simplify multiplication. = 12 Simplify addition.

In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.

• ⓔ 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 = 7 ( 15 ) − 2 [ ( 3 ) − 4 2 ] + 1 Simplify inside parentheses. = 7 ( 15 ) − 2 ( 3 − 16 ) + 1 Simplify exponent. = 7 ( 15 ) − 2 ( −13 ) + 1 Subtract. = 105 + 26 + 1 Multiply. = 132 Add. 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 = 7 ( 15 ) − 2 [ ( 3 ) − 4 2 ] + 1 Simplify inside parentheses. = 7 ( 15 ) − 2 ( 3 − 16 ) + 1 Simplify exponent. = 7 ( 15 ) − 2 ( −13 ) + 1 Subtract. = 105 + 26 + 1 Multiply. = 132 Add.
• ⓐ 5 2 − 4 2 + 7 ( 5 − 4 ) 2 5 2 − 4 2 + 7 ( 5 − 4 ) 2
• ⓑ 1 + 7 ⋅ 5 − 8 ⋅ 4 9 − 6 1 + 7 ⋅ 5 − 8 ⋅ 4 9 − 6
• ⓒ | 1.8 − 4.3 | + 0.4 15 + 10 | 1.8 − 4.3 | + 0.4 15 + 10
• ⓓ 1 2 [ 5 ⋅ 3 2 − 7 2 ] + 1 3 ⋅ 9 2 1 2 [ 5 ⋅ 3 2 − 7 2 ] + 1 3 ⋅ 9 2
• ⓔ [ ( 3 − 8 ) 2 − 4 ] − ( 3 − 8 ) [ ( 3 − 8 ) 2 − 4 ] − ( 3 − 8 )

Using Properties of Real Numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

Commutative Properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.

We can better see this relationship when using real numbers.

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

Again, consider an example with real numbers.

It is important to note that neither subtraction nor division is commutative. For example, 17 − 5 17 − 5 is not the same as 5 − 17. 5 − 17. Similarly, 20 ÷ 5 ≠ 5 ÷ 20. 20 ÷ 5 ≠ 5 ÷ 20.

Associative Properties

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

Consider this example.

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.

This property can be especially helpful when dealing with negative integers. Consider this example.

Are subtraction and division associative? Review these examples.

As we can see, neither subtraction nor division is associative.

Distributive Property

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.

This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.

Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.

To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.

A special case of the distributive property occurs when a sum of terms is subtracted.

For example, consider the difference 12 − ( 5 + 3 ) . 12 − ( 5 + 3 ) . We can rewrite the difference of the two terms 12 and ( 5 + 3 ) ( 5 + 3 ) by turning the subtraction expression into addition of the opposite. So instead of subtracting ( 5 + 3 ) , ( 5 + 3 ) , we add the opposite.

Now, distribute −1 −1 and simplify the result.

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

Identity Properties

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.

The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.

For example, we have ( −6 ) + 0 = −6 ( −6 ) + 0 = −6 and 23 ⋅ 1 = 23. 23 ⋅ 1 = 23. There are no exceptions for these properties; they work for every real number, including 0 and 1.

Inverse Properties

The inverse property of addition states that, for every real number a , there is a unique number, called the additive inverse (or opposite), denoted by (− a ), that, when added to the original number, results in the additive identity, 0.

For example, if a = −8 , a = −8 , the additive inverse is 8, since ( −8 ) + 8 = 0. ( −8 ) + 8 = 0.

The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a , there is a unique number, called the multiplicative inverse (or reciprocal), denoted 1 a , 1 a , that, when multiplied by the original number, results in the multiplicative identity, 1.

For example, if a = − 2 3 , a = − 2 3 , the reciprocal, denoted 1 a , 1 a , is − 3 2 − 3 2 because

Properties of Real Numbers

The following properties hold for real numbers a , b , and c .

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

• ⓐ 3 ⋅ 6 + 3 ⋅ 4 3 ⋅ 6 + 3 ⋅ 4
• ⓑ ( 5 + 8 ) + ( −8 ) ( 5 + 8 ) + ( −8 )
• ⓒ 6 − ( 15 + 9 ) 6 − ( 15 + 9 )
• ⓓ 4 7 ⋅ ( 2 3 ⋅ 7 4 ) 4 7 ⋅ ( 2 3 ⋅ 7 4 )
• ⓔ 100 ⋅ [ 0.75 + ( −2.38 ) ] 100 ⋅ [ 0.75 + ( −2.38 ) ]
• ⓐ 3 ⋅ 6 + 3 ⋅ 4 = 3 ⋅ ( 6 + 4 ) Distributive property. = 3 ⋅ 10 Simplify. = 30 Simplify. 3 ⋅ 6 + 3 ⋅ 4 = 3 ⋅ ( 6 + 4 ) Distributive property. = 3 ⋅ 10 Simplify. = 30 Simplify.
• ⓑ ( 5 + 8 ) + ( −8 ) = 5 + [ 8 + ( −8 ) ] Associative property of addition. = 5 + 0 Inverse property of addition. = 5 Identity property of addition. ( 5 + 8 ) + ( −8 ) = 5 + [ 8 + ( −8 ) ] Associative property of addition. = 5 + 0 Inverse property of addition. = 5 Identity property of addition.
• ⓒ 6 − ( 15 + 9 ) = 6 + [ ( −15 ) + ( −9 ) ] Distributive property. = 6 + ( −24 ) Simplify. = −18 Simplify. 6 − ( 15 + 9 ) = 6 + [ ( −15 ) + ( −9 ) ] Distributive property. = 6 + ( −24 ) Simplify. = −18 Simplify.
• ⓓ 4 7 ⋅ ( 2 3 ⋅ 7 4 ) = 4 7 ⋅ ( 7 4 ⋅ 2 3 ) Commutative property of multiplication. = ( 4 7 ⋅ 7 4 ) ⋅ 2 3 Associative property of multiplication. = 1 ⋅ 2 3 Inverse property of multiplication. = 2 3 Identity property of multiplication. 4 7 ⋅ ( 2 3 ⋅ 7 4 ) = 4 7 ⋅ ( 7 4 ⋅ 2 3 ) Commutative property of multiplication. = ( 4 7 ⋅ 7 4 ) ⋅ 2 3 Associative property of multiplication. = 1 ⋅ 2 3 Inverse property of multiplication. = 2 3 Identity property of multiplication.
• ⓔ 100 ⋅ [ 0.75 + ( − 2.38 ) ] = 100 ⋅ 0.75 + 100 ⋅ ( −2.38 ) Distributive property. = 75 + ( −238 ) Simplify. = −163 Simplify. 100 ⋅ [ 0.75 + ( − 2.38 ) ] = 100 ⋅ 0.75 + 100 ⋅ ( −2.38 ) Distributive property. = 75 + ( −238 ) Simplify. = −163 Simplify.
• ⓐ ( − 23 5 ) ⋅ [ 11 ⋅ ( − 5 23 ) ] ( − 23 5 ) ⋅ [ 11 ⋅ ( − 5 23 ) ]
• ⓑ 5 ⋅ ( 6.2 + 0.4 ) 5 ⋅ ( 6.2 + 0.4 )
• ⓒ 18 − ( 7 −15 ) 18 − ( 7 −15 )
• ⓓ 17 18 + [ 4 9 + ( − 17 18 ) ] 17 18 + [ 4 9 + ( − 17 18 ) ]
• ⓔ 6 ⋅ ( −3 ) + 6 ⋅ 3 6 ⋅ ( −3 ) + 6 ⋅ 3

Evaluating Algebraic Expressions

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as x + 5 , 4 3 π r 3 , x + 5 , 4 3 π r 3 , or 2 m 3 n 2 . 2 m 3 n 2 . In the expression x + 5 , x + 5 , 5 is called a constant because it does not vary and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

Describing Algebraic Expressions

List the constants and variables for each algebraic expression.

• ⓑ 4 3 π r 3 4 3 π r 3
• ⓒ 2 m 3 n 2 2 m 3 n 2
• ⓐ 2 π r ( r + h ) 2 π r ( r + h )
• ⓑ 2( L + W )
• ⓒ 4 y 3 + y 4 y 3 + y

Evaluating an Algebraic Expression at Different Values

Evaluate the expression 2 x − 7 2 x − 7 for each value for x.

• ⓐ x = 0 x = 0
• ⓑ x = 1 x = 1
• ⓒ x = 1 2 x = 1 2
• ⓓ x = −4 x = −4
• ⓐ Substitute 0 for x . x . 2 x − 7 = 2 ( 0 ) − 7 = 0 − 7 = −7 2 x − 7 = 2 ( 0 ) − 7 = 0 − 7 = −7
• ⓑ Substitute 1 for x . x . 2 x − 7 = 2 ( 1 ) − 7 = 2 − 7 = −5 2 x − 7 = 2 ( 1 ) − 7 = 2 − 7 = −5
• ⓒ Substitute 1 2 1 2 for x . x . 2 x − 7 = 2 ( 1 2 ) − 7 = 1 − 7 = −6 2 x − 7 = 2 ( 1 2 ) − 7 = 1 − 7 = −6
• ⓓ Substitute −4 −4 for x . x . 2 x − 7 = 2 ( − 4 ) − 7 = − 8 − 7 = −15 2 x − 7 = 2 ( − 4 ) − 7 = − 8 − 7 = −15

Evaluate the expression 11 − 3 y 11 − 3 y for each value for y.

• ⓐ y = 2 y = 2
• ⓑ y = 0 y = 0
• ⓒ y = 2 3 y = 2 3
• ⓓ y = −5 y = −5

Evaluate each expression for the given values.

• ⓐ x + 5 x + 5 for x = −5 x = −5
• ⓑ t 2 t −1 t 2 t −1 for t = 10 t = 10
• ⓒ 4 3 π r 3 4 3 π r 3 for r = 5 r = 5
• ⓓ a + a b + b a + a b + b for a = 11 , b = −8 a = 11 , b = −8
• ⓔ 2 m 3 n 2 2 m 3 n 2 for m = 2 , n = 3 m = 2 , n = 3
• ⓐ Substitute −5 −5 for x . x . x + 5 = ( −5 ) + 5 = 0 x + 5 = ( −5 ) + 5 = 0
• ⓑ Substitute 10 for t . t . t 2 t − 1 = ( 10 ) 2 ( 10 ) − 1 = 10 20 − 1 = 10 19 t 2 t − 1 = ( 10 ) 2 ( 10 ) − 1 = 10 20 − 1 = 10 19
• ⓒ Substitute 5 for r . r . 4 3 π r 3 = 4 3 π ( 5 ) 3 = 4 3 π ( 125 ) = 500 3 π 4 3 π r 3 = 4 3 π ( 5 ) 3 = 4 3 π ( 125 ) = 500 3 π
• ⓓ Substitute 11 for a a and –8 for b . b . a + a b + b = ( 11 ) + ( 11 ) ( −8 ) + ( −8 ) = 11 − 88 − 8 = −85 a + a b + b = ( 11 ) + ( 11 ) ( −8 ) + ( −8 ) = 11 − 88 − 8 = −85
• ⓔ Substitute 2 for m m and 3 for n . n . 2 m 3 n 2 = 2 ( 2 ) 3 ( 3 ) 2 = 2 ( 8 ) ( 9 ) = 144 = 12 2 m 3 n 2 = 2 ( 2 ) 3 ( 3 ) 2 = 2 ( 8 ) ( 9 ) = 144 = 12
• ⓐ y + 3 y − 3 y + 3 y − 3 for y = 5 y = 5
• ⓑ 7 − 2 t 7 − 2 t for t = −2 t = −2
• ⓒ 1 3 π r 2 1 3 π r 2 for r = 11 r = 11
• ⓓ ( p 2 q ) 3 ( p 2 q ) 3 for p = −2 , q = 3 p = −2 , q = 3
• ⓔ 4 ( m − n ) − 5 ( n − m ) 4 ( m − n ) − 5 ( n − m ) for m = 2 3 , n = 1 3 m = 2 3 , n = 1 3

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation 2 x + 1 = 7 2 x + 1 = 7 has the solution of 3 because when we substitute 3 for x x in the equation, we obtain the true statement 2 ( 3 ) + 1 = 7. 2 ( 3 ) + 1 = 7.

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area A A of a circle in terms of the radius r r of the circle: A = π r 2 . A = π r 2 . For any value of r , r , the area A A can be found by evaluating the expression π r 2 . π r 2 .

Using a Formula

A right circular cylinder with radius r r and height h h has the surface area S S (in square units) given by the formula S = 2 π r ( r + h ) . S = 2 π r ( r + h ) . See Figure 3 . Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of π . π .

Evaluate the expression 2 π r ( r + h ) 2 π r ( r + h ) for r = 6 r = 6 and h = 9. h = 9.

The surface area is 180 π 180 π square inches.

A photograph with length L and width W is placed in a mat of width 8 centimeters (cm). The area of the mat (in square centimeters, or cm 2 ) is found to be A = ( L + 16 ) ( W + 16 ) − L ⋅ W . A = ( L + 16 ) ( W + 16 ) − L ⋅ W . See Figure 4 . Find the area of a mat for a photograph with length 32 cm and width 24 cm.

Simplifying Algebraic Expressions

Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

Simplify each algebraic expression.

• ⓐ 3 x − 2 y + x − 3 y − 7 3 x − 2 y + x − 3 y − 7
• ⓑ 2 r − 5 ( 3 − r ) + 4 2 r − 5 ( 3 − r ) + 4
• ⓒ ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) ( 4 t − 5 4 s ) − ( 2 3 t + 2 s )
• ⓓ 2 m n − 5 m + 3 m n + n 2 m n − 5 m + 3 m n + n
• ⓐ 3 x − 2 y + x − 3 y − 7 = 3 x + x − 2 y − 3 y − 7 Commutative property of addition. = 4 x − 5 y − 7 Simplify. 3 x − 2 y + x − 3 y − 7 = 3 x + x − 2 y − 3 y − 7 Commutative property of addition. = 4 x − 5 y − 7 Simplify.
• ⓑ 2 r − 5 ( 3 − r ) + 4 = 2 r − 15 + 5 r + 4 Distributive property. = 2 r + 5 r − 15 + 4 Commutative property of addition. = 7 r − 11 Simplify. 2 r − 5 ( 3 − r ) + 4 = 2 r − 15 + 5 r + 4 Distributive property. = 2 r + 5 r − 15 + 4 Commutative property of addition. = 7 r − 11 Simplify.
• ⓒ ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) = 4 t − 5 4 s − 2 3 t − 2 s Distributive property. = 4 t − 2 3 t − 5 4 s − 2 s Commutative property of addition. = 10 3 t − 13 4 s Simplify. ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) = 4 t − 5 4 s − 2 3 t − 2 s Distributive property. = 4 t − 2 3 t − 5 4 s − 2 s Commutative property of addition. = 10 3 t − 13 4 s Simplify.
• ⓓ 2 m n − 5 m + 3 m n + n = 2 m n + 3 m n − 5 m + n Commutative property of addition. = 5 m n − 5 m + n Simplify. 2 m n − 5 m + 3 m n + n = 2 m n + 3 m n − 5 m + n Commutative property of addition. = 5 m n − 5 m + n Simplify.
• ⓐ 2 3 y − 2 ( 4 3 y + z ) 2 3 y − 2 ( 4 3 y + z )
• ⓑ 5 t − 2 − 3 t + 1 5 t − 2 − 3 t + 1
• ⓒ 4 p ( q − 1 ) + q ( 1 − p ) 4 p ( q − 1 ) + q ( 1 − p )
• ⓓ 9 r − ( s + 2 r ) + ( 6 − s ) 9 r − ( s + 2 r ) + ( 6 − s )

Simplifying a Formula

A rectangle with length L L and width W W has a perimeter P P given by P = L + W + L + W . P = L + W + L + W . Simplify this expression.

If the amount P P is deposited into an account paying simple interest r r for time t , t , the total value of the deposit A A is given by A = P + P r t . A = P + P r t . Simplify the expression. (This formula will be explored in more detail later in the course.)

Access these online resources for additional instruction and practice with real numbers.

• Simplify an Expression.
• Evaluate an Expression 1.
• Evaluate an Expression 2.

1.1 Section Exercises

Is 2 2 an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?

What do the Associative Properties allow us to do when following the order of operations? Explain your answer.

For the following exercises, simplify the given expression.

10 + 2 × ( 5 − 3 ) 10 + 2 × ( 5 − 3 )

6 ÷ 2 − ( 81 ÷ 3 2 ) 6 ÷ 2 − ( 81 ÷ 3 2 )

18 + ( 6 − 8 ) 3 18 + ( 6 − 8 ) 3

−2 × [ 16 ÷ ( 8 − 4 ) 2 ] 2 −2 × [ 16 ÷ ( 8 − 4 ) 2 ] 2

4 − 6 + 2 × 7 4 − 6 + 2 × 7

3 ( 5 − 8 ) 3 ( 5 − 8 )

4 + 6 − 10 ÷ 2 4 + 6 − 10 ÷ 2

12 ÷ ( 36 ÷ 9 ) + 6 12 ÷ ( 36 ÷ 9 ) + 6

( 4 + 5 ) 2 ÷ 3 ( 4 + 5 ) 2 ÷ 3

3 − 12 × 2 + 19 3 − 12 × 2 + 19

2 + 8 × 7 ÷ 4 2 + 8 × 7 ÷ 4

5 + ( 6 + 4 ) − 11 5 + ( 6 + 4 ) − 11

9 − 18 ÷ 3 2 9 − 18 ÷ 3 2

14 × 3 ÷ 7 − 6 14 × 3 ÷ 7 − 6

9 − ( 3 + 11 ) × 2 9 − ( 3 + 11 ) × 2

6 + 2 × 2 − 1 6 + 2 × 2 − 1

64 ÷ ( 8 + 4 × 2 ) 64 ÷ ( 8 + 4 × 2 )

9 + 4 ( 2 2 ) 9 + 4 ( 2 2 )

( 12 ÷ 3 × 3 ) 2 ( 12 ÷ 3 × 3 ) 2

25 ÷ 5 2 − 7 25 ÷ 5 2 − 7

( 15 − 7 ) × ( 3 − 7 ) ( 15 − 7 ) × ( 3 − 7 )

2 × 4 − 9 ( −1 ) 2 × 4 − 9 ( −1 )

4 2 − 25 × 1 5 4 2 − 25 × 1 5

12 ( 3 − 1 ) ÷ 6 12 ( 3 − 1 ) ÷ 6

For the following exercises, evaluate the expression using the given value of the variable.

8 ( x + 3 ) – 64 8 ( x + 3 ) – 64 for x = 2 x = 2

4 y + 8 – 2 y 4 y + 8 – 2 y for y = 3 y = 3

( 11 a + 3 ) − 18 a + 4 ( 11 a + 3 ) − 18 a + 4 for a = –2 a = –2

4 z − 2 z ( 1 + 4 ) – 36 4 z − 2 z ( 1 + 4 ) – 36 for z = 5 z = 5

4 y ( 7 − 2 ) 2 + 200 4 y ( 7 − 2 ) 2 + 200 for y = –2 y = –2

− ( 2 x ) 2 + 1 + 3 − ( 2 x ) 2 + 1 + 3 for x = 2 x = 2

For the 8 ( 2 + 4 ) − 15 b + b 8 ( 2 + 4 ) − 15 b + b for b = –3 b = –3

2 ( 11 c − 4 ) – 36 2 ( 11 c − 4 ) – 36 for c = 0 c = 0

4 ( 3 − 1 ) x – 4 4 ( 3 − 1 ) x – 4 for x = 10 x = 10

1 4 ( 8 w − 4 2 ) 1 4 ( 8 w − 4 2 ) for w = 1 w = 1

For the following exercises, simplify the expression.

4 x + x ( 13 − 7 ) 4 x + x ( 13 − 7 )

2 y − ( 4 ) 2 y − 11 2 y − ( 4 ) 2 y − 11

a 2 3 ( 64 ) − 12 a ÷ 6 a 2 3 ( 64 ) − 12 a ÷ 6

8 b − 4 b ( 3 ) + 1 8 b − 4 b ( 3 ) + 1

5 l ÷ 3 l × ( 9 − 6 ) 5 l ÷ 3 l × ( 9 − 6 )

7 z − 3 + z × 6 2 7 z − 3 + z × 6 2

4 × 3 + 18 x ÷ 9 − 12 4 × 3 + 18 x ÷ 9 − 12

9 ( y + 8 ) − 27 9 ( y + 8 ) − 27

( 9 6 t − 4 ) 2 ( 9 6 t − 4 ) 2

6 + 12 b − 3 × 6 b 6 + 12 b − 3 × 6 b

18 y − 2 ( 1 + 7 y ) 18 y − 2 ( 1 + 7 y )

( 4 9 ) 2 × 27 x ( 4 9 ) 2 × 27 x

8 ( 3 − m ) + 1 ( − 8 ) 8 ( 3 − m ) + 1 ( − 8 )

9 x + 4 x ( 2 + 3 ) − 4 ( 2 x + 3 x ) 9 x + 4 x ( 2 + 3 ) − 4 ( 2 x + 3 x )

5 2 − 4 ( 3 x ) 5 2 − 4 ( 3 x )

Real-World Applications

For the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $10 on a streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor’s dog.

Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.

How much money does Fred keep?

For the following exercises, solve the given problem.

According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by π . π . Is the circumference of a quarter a whole number, a rational number, or an irrational number?

Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact?

For the following exercises, consider this scenario: There is a mound of g g pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.

Write the equation that describes the situation.

Solve for g .

For the following exercise, solve the given problem.

Ramon runs the marketing department at their company. Their department gets a budget every year, and every year, they must spend the entire budget without going over. If they spend less than the budget, then the department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. They must spend the budget such that 2,500,000 − x = 0. 2,500,000 − x = 0. What property of addition tells us what the value of x must be?

For the following exercises, use a graphing calculator to solve for x . Round the answers to the nearest hundredth.

0.5 ( 12.3 ) 2 − 48 x = 3 5 0.5 ( 12.3 ) 2 − 48 x = 3 5

( 0.25 − 0.75 ) 2 x − 7.2 = 9.9 ( 0.25 − 0.75 ) 2 x − 7.2 = 9.9

If a whole number is not a natural number, what must the number be?

Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.

Determine whether the statement is true or false: The product of a rational and irrational number is always irrational.

Determine whether the simplified expression is rational or irrational: −18 − 4 ( 5 ) ( −1 ) . −18 − 4 ( 5 ) ( −1 ) .

Determine whether the simplified expression is rational or irrational: −16 + 4 ( 5 ) + 5 . −16 + 4 ( 5 ) + 5 .

The division of two natural numbers will always result in what type of number?

What property of real numbers would simplify the following expression: 4 + 7 ( x − 1 ) ? 4 + 7 ( x − 1 ) ?

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Rational and Irrational Numbers

Hello, and welcome to this video about rational and irrational numbers, which are key components of the real number system.

Today, we’re going to explore these two subsets of real numbers that you encounter every day, often without even realizing it. Rational numbers are the familiar ones you use in daily life, like 10 dollars or ¾ of a cup. Then, there are the irrational numbers—less obvious but equally fascinating, like the square root of 2 or the constant \(\pi\).

What are Real Numbers?

This Venn diagram is a visual representation of how real numbers are classified.

You can see that rational numbers include natural numbers, whole numbers, and integers. Natural numbers comprise the smallest subset, which is also known as the set of “counting” numbers. These are all positive, non-decimal values starting at one. Whole numbers encompass all natural numbers, with the addition of zero. Integers are whole numbers and their additive inverses (negatives).

What are Rational Numbers?

Rational numbers include all of the sets seen here in addition to some values in between.

An easy way to remember this is that the word ratio is in the name of this classification. All numbers included in the rational number set can be written as a ratio of integers.

Rational numbers are any numbers that can be written as \(\frac{a}{b}\) , as long as \(a\) and \(b\) are integers and \(b≠0\) .

For example, the integer 3 could be represented as the fractions \(\frac{3}{1}\) , \(\frac{6}{2}\) , or even \(\frac{-24}{-8}\) .

The integer 0 could be represented as the fractions \(\frac{0}{3}\) , \(\frac{0}{-2}\) , or \(\frac{0}{123}\) .

Fractions can also be written as decimals. For example, \(\frac{13}{100}\) is equal to 0.13 because the 3 is in the hundredths decimal place and the 1 is in the tenths decimal place. This is an example of a terminating decimal.

Other decimals have repeating patterns. These are considered rational because they can be expressed as a fraction. For example, the repeating decimal \(2.1 \overline7\) represents the digits 2.17171717… and so on. This can be represented as the fraction \(\frac{215}{99}\) .

Irrational Number Examples

It is important to note that not all decimals are repeating. Some decimals have an infinite number of non-repeating digits. These types of real numbers cannot be expressed as a ratio of integers and are therefore classified as irrational.

While there are an infinite number of irrational numbers in the real number system, those most commonly used in mathematics are the square roots of non-perfect squares, like the \(\sqrt2\) , for example, and the constants \(\pi\) and Euler’s number (\(e\)). The notation for irrational numbers allows for efficiency in mathematical applications.

Let’s take a look at some example problems involving rational and irrational numbers.

Determine if the following numbers are rational or irrational and explain your reasoning.

• \(\sqrt 7\)
• \(\frac{1}{3}\)

(A) is rational because -2 is an integer, which is a subset of rational numbers. (B) is rational because 0 is a whole number, which is also a subset of rational numbers. (C) is irrational because the square root of 7 is approximately equal to 2.6457513, but is an infinitely non-repeating decimal. (D) is rational because \(\frac{1}{3}\) is equal to \(0.\overline3\) . Any repeating decimal or number that can be written as a fraction of integers is a rational number.

Here’s another example: List 3 rational numbers between 3 and 4. For this example, let’s stick with decimals.

We can keep it simple and do 3.25, 3.5, and 3.75, but in reality any terminating decimal like 3.58 or 3.987 would work in this scenario. Remember, any number between 3 and 4 that can be written as a fraction would also be correct.

All right, that’s all for this review. Thanks for watching, and happy studying!

Frequently Asked Questions

Are all integers rational numbers.

Yes, a rational number is any number that can be expressed as a fraction. All integers fit this definition.

Are negative numbers rational?

Yes, most negative numbers are rational. A rational number is any number that can be written as a fraction. These include whole numbers, fractions, decimals that end, and decimals that repeat. Positive and negative do not affect rationality.

Are all rational numbers whole numbers?

No, not all rational numbers are whole numbers. Rational numbers include all numbers that end or repeat. A whole number is any number without a fractional part that is greater than or equal to zero. Ex. 2.7 is a rational number but not a whole number.

What is the difference between rational and irrational numbers?

The difference between rational and irrational numbers is that a rational number can be represented as an exact fraction and an irrational number cannot. A rational number includes any whole number, fraction, or decimal that ends or repeats. An irrational number is any number that cannot be turned into a fraction, so any number that does not fit the definition of a rational number.

Rational and Irrational Number Practice Questions

  Is π rational?

Cannot be determined

The correct answer is no. Pi (π) is an irrational number because it is a never-ending decimal that cannot be simplified as an exact fraction.

  Is \(1.\overline{3}\) a rational number?

The correct answer is yes. \(1.\overline{3}\) can be represented as the fraction \(1\frac{1}{3}\), which means it is rational. Any number that can be represented as a fraction is considered rational.

  Which of the following numbers is an example of a rational number?

The correct answer is 4.17. This is the only number out of this list that can be turned into a fraction, \(4\frac{17}{100}\).

  Which of the following is an irrational number?

The correct answer is \(\sqrt{3}\). Square roots of non-perfect squares are not rational because they are equal to a never-ending decimal number, which means it is a number that cannot be turned into a fraction.

  Is \(\frac{7}{9}\) rational?

The correct answer is yes. A rational number is any number that can be turned into a fraction, and \(\frac{7}{9}\) is a fraction.

Return to Basic Arithmetic Videos

by Mometrix Test Preparation | This Page Last Updated: April 1, 2024

Real Number System Activities

By: Author Sarah Carter

Posted on Published: November 27, 2023  - Last updated: April 24, 2024

Categories Real Number System

Looking for a fun hands-on real number system activity to help students learn the difference between integers and irrational numbers?

Activities for Visualizing the Real Numbers

Real number system nesting boxes.

Build a set of DIY nesting boxes to help illustrate the real number system to students. When the integer box fits inside the rational number box, students will be able to visually see why every integer is a rational number but every rational number is not an integer.

Real Number System Graphic Organizer

This graphic organizer attempts to capture the same illustration of the real number system as the nesting boxes with a 2D diagram.

Real Number System Projects

Real number line project.

Students must construct a number line with a specified number of natural numbers, integers that are not whole numbers,, irrational numbers, and rational numbers that are not integers.

Activities for Classifying Real Numbers

Real number system card activity.

I created this set of real number system cards several years ago to give my students much-needed practice determining which subsets of the real number system a number belongs to.

Always Sometimes Never Dice Activity for Real Number System

I created this always sometimes never dice activity for classifying numbers according to the real number system. We completed this after sorting numbers using our Real Number System Nesting Boxes. I always hated Always, Sometimes, Never questions when I did them as a geometry student in high school. But, I decided to give it a try with my students.

Subsets of the Real Numbers Graphic Organizer

Use this subsets of the real numbers graphic organizer to help students keep the definitions for natural numbers, whole numbers, integers, rational numbers, and irrational numbers organized.

Rational vs Irrational Numbers

Rational and irrational numbers exploration activity.

This exploration activity will get students thinking about what happens when you perform various operations with rational and irrational numbers.

More Activities for Teaching the Real Number System

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Real Number System Unit 8th Grade CCSS

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An 8 day CCSS-Aligned Real Number System Unit includes squares and square roots, rational vs. irrational numbers, classifying real numbers, and comparing and ordering real numbers.

Students will practice with both skill-based problems, real-world application questions, and error analysis to support higher level thinking skills.  You can reach your students and teach the standards without all of the prep and stress of creating materials!

Standards: 8.NS.1, 8.NS.2, 8.EE.2;  Texas Teacher?  Grab the TEKS-Aligned Real Number System Unit.  Please don’t purchase both as there is overlapping content.

Learning Focus:

• approximate the value of an irrational number and locate the value on a number line
• classify, compare, and order real numbers
• convert between fractions and decimals and evaluate square roots

what is included in the 8th grade ccss real number system unit?

1. Unit Overviews

• Streamline planning with unit overviews that include essential questions, big ideas, vertical alignment, vocabulary, and common misconceptions.
• A pacing guide and tips for teaching each topic are included to help you be more efficient in your planning.

2. Student Handouts

• Student-friendly guided notes are scaffolded to support student learning.
• Available as a PDF and the student handouts/homework/study guides have been converted to Google Slides™ for your convenience.

3. Independent Practice

• Daily homework is aligned directly to the student handouts and is versatile for both in class or at home practice.

4. Assessments

• 1-2 quizzes, a unit study guide, and a unit test allow you to easily assess and meet the needs of your students.
• The Unit Test is available as an editable PPT, so that you can modify and adjust questions as needed.

5. Answer Keys

• All answer keys are included.

***Please download a preview to see sample pages and more information.***

How to use this resource:

• Use as a whole group, guided notes setting
• Use in a small group, math workshop setting
• Chunk each student handout to incorporate whole group instruction, small group practice, and independent practice.
• Incorporate our  Real Number System Activity Bundle  for hands-on activities as additional and engaging practice opportunities.

Time to Complete:

• Each student handout is designed for a single class period. However, feel free to review the problems and select specific ones to meet your student needs. There are multiple problems to practice the same concepts, so you can adjust as needed.

Is this resource editable?

• The unit test is editable with Microsoft PPT. The remainder of the file is a PDF and not editable.

Looking for more 8th Grade Math Material? Join our All Access Membership Community! You can reach your students without the “I still have to prep for tomorrow” stress, the constant overwhelm of teaching multiple preps, and the hamster wheel demands of creating your own teaching materials.

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This resource is often paired with:

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Real Number System Unit 8th Grade TEKS

Rational vs irrational numbers

Learning the difference between rational and irrational numbers doesn’t have to be complicated, let’s find out how together.

Author Amber Watkins

Published August 2, 2023

Rational vs irrational numbers?

• Key takeaways
• Rational numbers can be written as fractions and ratios.
• The most common rational numbers are positive and negative integers.
• The most common irrational number used in math is Pi.

Table of contents

• Practice problems

Rational numbers are the most common numbers we see in the world around us. We use rational numbers on our speed limit signs, in our recipes, and on our shoe labels to show what size we wear.

Irrational numbers are not as common, but they are very important! The most common irrational number is Pi! Without that very long, never-ending number we wouldn’t be able to calculate the area of a circle .

Examples of rational and irrational numbers

Rational numbers can be defined as any number or value that can be written as a fraction or a ratio. Any number that can not be written as a fraction is irrational. Let’s review some examples to help us identify rational vs irrational numbers .

Rational number examples

1. Integers: all integers are rational numbers . Integers include all real numbers both positive and negative . Since all integers can be written as a fraction, they are all rational numbers.

Example: Is 3 a rational number?

3 can be written as 3/1 or 6/2.

Since it can be written as a ratio and fraction, it is a rational number.

2. Repeating decimals: All repeating decimals are rational numbers. Repeating decimals have numbers after the decimal place that repeat. Even though it may be difficult to change repeating decimals into fractions, it is possible, so they are considered rational numbers.

Example: Is .66666666 a rational number?

.66666666 can be written as ⅔.

3. Non-repeating decimals that are finite: All decimals that are finite (come to an end) are rational numbers. Finite decimals can be written as fractions, so they are rational numbers.

Example: is 5.25 a rational number?

5.25 can be written as 525/100.

4. Perfect square roots: All perfect square roots are rational numbers . Perfect square roots always have a whole number as the answer. Since the square roots are positive whole numbers, they are rational and therefore make the square root rational.

Example: Is √25 a rational number?

The square root of 25 is 5. Five can be written as 5/1.

Irrational number examples

1. Non-repeating, non-terminitating decimals: All decimals that do no repeat and continue indefinitely are the most common irrational numbers. Decimals that never end can not be written as a fraction without rounding. For this reason they are considered to be irrational, not rational.

Example: Is 5.432698762 a rational number?

No, it is an irrational number because it can’t be written as a fraction.

Did you know?

The first 12 digits of Pi are 3.14159265358 and it continues on forever.

2. Non-perfect square roots: All square roots that are not perfect squares are classified as irrational. You can enter the square root of the numbers 7, 12, or 18 into a calculator, but the answer you get will not be rational. This is an indication that the square root of 7, 12, and 18 are irrational numbers.

Example: Is √12 a rational number?

The √12 is 3.4610161514 which can not be made into a fraction.

No, it is an irrational number.

Difference between rational and irrational numbers

As you can see from the examples, the primary difference between rational and irrational numbers is rational numbers can be written as fractions, irrational numbers can not. Numbers in the form of decimals and square roots can be classified as rational and irrational numbers , so we have to be extra careful when checking. Now it’s your turn to practice!

Explore numbers with DoodleMath

Want to learn more about rational and irrational numbers? DoodleMath is an award-winning math app that’s proven to double a child’s rate of progression with just 10 minutes of use a day!

Filled with fun, interactive questions aligned to state standards, Doodle creates a unique work program tailored to each child’s needs, boosting their confidence and skills in math. Try it free  today!

Practice problems: rational vs irrational numbers

.25 is a rational number because it can be written as the fraction 1/4

√100 is a rational number because the square root of 100 is 10 and 10 can be written as 10/1 so it is a rational number, so √100 is also a rational number

4.986432 is an irrational number because it is a non-repeating decimal and it can’t be put in the form of a fraction or ratio

FAQs about rational and irrational numbers

We know learning the difference between rational and irrational numbers is complex so we’ve provided a few frequently asked questions many students or parents have when they start working with these numbers.

The primary difference between rational numbers and irrational numbers is whether the numbers can be written as fractions. In order to determine whether a number is rational or irrational, you must check to see if the number can be written as a fraction.

The most common irrational number is Pi or 3.14159265358. Some examples of rational numbers are 5, 10, 3/4, and .80. For more examples of rational and irrational numbers see our math practice app.

Related Posts

About Rational Numbers

Irrational Numbers

Composite Numbers

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Amber Watkins

Amber is an education specialist with a degree in Early Childhood Education. She has over 12 years of experience teaching and tutoring elementary through college level math. "Knowing that my work in math education makes such an impact leaves me with an indescribable feeling of pride and joy!"

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Real Number System Lesson 4 Rational vs. Irrational Numbers

• Google Slides™

Description

This lesson is from my 8th grade Real Number System Unit.

This lesson teaches the difference between rational and irrational numbers.

This unit is aligned with standards KY.8.NS.1 and KY.8.NS.2.

Questions & Answers

Ms grace middle school math.

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Number Systems

There are a variety of number systems, a handful of which are used on a regular basis for basic mathematics in intermediate and high school. These include natural numbers , integers , rational numbers , irrational numbers , real numbers , and more. Continue reading to learn more about the properties of each of these types of numbers.

• Natural numbers

The natural (or counting) numbers are the part of the number system that includes all the positive integers from 1 through infinity. They are used for the purpose of counting. Natural numbers do not include 0, fractions , decimals , or negative numbers.

The set of natural numbers is usually represented by the letter "N".

N = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 … }

Natural numbers include all whole numbers except for the number 0. In other words, all natural numbers are whole numbers, but not all whole numbers are natural numbers.

There are four properties that natural numbers fit into:

• Closure property – the sum or product of any two natural numbers is a natural number. This does not hold true for subtraction or division .
• Commutative property – the order in which you add or multiply natural numbers does not affect the result. This does not hold true for subtraction or division.
• Associative property – the way natural numbers are grouped in addition or multiplication does not affect the result. This does not hold true for subtraction or division.
• Distributive property – multiplication of natural numbers is always distributive over addition.

The integers are a set of numbers consisting of the natural numbers, their additive inverses, and zero. In other words, they include all positive numbers, negative numbers, and 0. They can never be a fraction or decimal. All natural numbers are integers that start from 1 and end at infinity. All whole numbers are integers that start from 0 and end at infinity.

The set of integers is usually represented by the letter "Z". It can also be represented by the letter "J".

Z = { … -3 , -2 , -1 , 0 , 1 , 2 , 3 , 4 … }

The sum, product, and difference of any two integers is always an integer. The same is not true for division.

There are four types of numbers that fall under the integer category.

• Whole numbers
• Odd and even integers
• Prime and composite numbers

There are rules for integers based on the four basic operations:

Addition rule – If the sign of both integers is the same, the result will have the same sign. Two positives equal a positive and two negatives equal a negative.

- 14 + ( - 12 ) = - 26

If the two numbers being added have a different sign, it will lead to a subtraction and the result will have the sign of the larger (in absolute value) integer.

- 2 + 10 = 8

- 10 + 2 = 8

Subtraction rule – Keep the sign of the first number the same, change the operator from subtraction to addition, and change the sign of the second number. Once you have applied this rule, follow the rules for adding integers.

Multiplication division rule – If the signs are the same, multiply or divide, and the answer is always positive.

5 ∗ 5 = 25

-5 ∗ ( - 5 ) = 25

If the signs are different, multiply or divide, and the answer is always negative.

-5 ∗ 5 = -25

25 - 5 = - 5

There are five properties that integers fit into. The first four are the same as natural numbers. The last one is the identity property.

The additive identity property states that any integer added to 0 will give the same number. So 0 is called the additive identity. For any integer x,

x + 0 = x = 0 + x

The multiplicative identity states that when an integer is multiplied by 1, it will give the integer itself as the product. So 1 is called the multiplicative identity. For any integer x,

x ∗ 1 = x = 1 ∗ x

If any integer is multiplied by 0, the product will be 0.

x ∗ 0 = 0 = 0 ∗ x

If any integer is multiplied by -1, the product will be the opposite of the number.

x ∗ - 1 = - x = -1 ∗ x

Rational numbers

Rational numbers can be expressed as a ratio between two integers. For example, the fractions 1 3 and -1211 19 are both rational numbers.

The rational numbers include all the integers because any integer z can be expressed as the ratio z 1 .

All decimals that terminate are also rational numbers because, for example, 8.27 can be expressed as 827 100 . Decimals that have a repeating pattern at some point are also rational. For example, 0.0833333 … = 1 12 .

Some of the important properties of the rational numbers are as follows:

• The results are always a rational number if you add, subtract, or multiply any two rational numbers.
• A rational number remains the same if you divide or multiply both the numerator and denominator with the same factor.
• If you add 0 to a rational number, the result will be the number itself.
• Rational numbers are closed under addition, subtraction, and multiplication.

Irrational numbers

An irrational number is one that cannot be written as a ratio, or fraction of integers. In decimal form, it never ends or repeats. The ancient Greeks discovered that not all numbers are rational. There are equations that cannot be solved by using ratios of integers.

One of the first such equations to be studied was 2 = x 2 . What number times itself equals 2?

2 is about 1.414 because 1.414 2 = 1.999396 , which is close to 2. But you'll never reach 2 exactly by squaring a fraction (or terminating decimal ). The square root of 2 is an irrational number, meaning its decimal equivalent goes on forever, with no repeating pattern.

2 = 1.41421356237309 …

Another famous irrational number is the golden ratio , a number that has great importance in biology.

1 + 5 2 = 1.61803398874989 …

Another famous irrational number is pi , the ratio of the circumference of a circle to its diameter.

π = 3.14159265358979 …

And another famous irrational number is e , the most important number in calculus.

e = 2.718281828455904 …

Irrational numbers can be further divided into algebraic numbers, which are the solutions of some polynomial equations (such as 2 and the golden ratio), and transcendental numbers , which are not the solutions of any polynomial equation. π and e are both transcendental.

Real numbers

The real numbers are the set of numbers that contain all of the rational numbers and all of the irrational numbers. The real numbers are "all the numbers" on the number line. There are infinitely many real numbers, just as there are infinitely many numbers in each of the other sets of numbers. But it can be proved that the infinity of the real numbers is a bigger infinity!

The smaller, or countable, infinity of the integers and rationals is sometimes called ℵ ⁡ 0 (aleph-naught), and the uncountable infinity of the reals is called ℵ ⁡ 1 or aleph-one.

There are actually even bigger infinities, but you would want to take a set theory class to learn about those.

Complex numbers

The complex numbers are the set { a + b i | a and b are real numbers}, where i is the imaginary unit - 1 .

The complex numbers include the set of real numbers. The real numbers, in the complex system, are written as a + 0 i = a where a is a real number.

This set is sometimes written as C for short. The set of complex numbers is important because for any polynomial p ( x ) with real number coefficients, all the solutions of p ( x ) = 0 will be in C.

..and beyond

There are even bigger sets of numbers that are used by mathematicians. For example, the hyper-real numbers or the quaternions, discovered by William H. Hamilton in 1845, form a number system with three different imaginary units!

Topics related to the Number Systems

Base (Number Systems)

Product of Powers Property

Flashcards covering the Number Systems

8th Grade Math Flashcards

Common Core: 8th Grade Math Flashcards

Practice tests covering the Number Systems

MAP 8th Grade Math Practice Tests

8th Grade Math Practice Tests

Get help learning about number systems

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Working with a math tutor helps your student in many different ways. A private tutor will move at your student's pace, taking the time your student needs to make sure they understand each concept thoroughly before moving on to the next concept. They are right there to answer questions as your student thinks of them so they don't waste time completing equations the wrong way. Contact the Educational Directors at Varsity Tutors to see how tutoring can help your student today.

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Unit 1: Rational & irrational numbers

Repeating decimals.

• Converting a fraction to a repeating decimal (Opens a modal)
• Converting repeating decimals to fractions (part 1 of 2) (Opens a modal)
• Converting repeating decimals to fractions (part 2 of 2) (Opens a modal)
• Writing repeating decimals as fractions review (Opens a modal)
• Writing fractions as repeating decimals review (Opens a modal)
• Writing fractions as repeating decimals Get 5 of 7 questions to level up!
• Converting repeating decimals to fractions Get 5 of 7 questions to level up!
• Converting multi-digit repeating decimals to fractions Get 3 of 4 questions to level up!

Square roots & cube roots

• Intro to square roots (Opens a modal)
• Square roots of perfect squares (Opens a modal)
• Intro to cube roots (Opens a modal)
• Worked example: Cube root of a negative number (Opens a modal)
• Square root of decimal (Opens a modal)
• Dimensions of a cube from its volume (Opens a modal)
• Square roots review (Opens a modal)
• Cube roots review (Opens a modal)
• Square roots Get 5 of 7 questions to level up!
• Cube roots Get 5 of 7 questions to level up!
• Equations with square roots & cube roots Get 3 of 4 questions to level up!
• Roots of decimals & fractions Get 3 of 4 questions to level up!
• Equations with square roots: decimals & fractions Get 3 of 4 questions to level up!
• Square and cube challenge Get 3 of 4 questions to level up!

Irrational numbers

• Intro to rational & irrational numbers (Opens a modal)
• Classifying numbers: rational & irrational (Opens a modal)
• Classifying numbers (Opens a modal)
• Classifying numbers review (Opens a modal)
• Worked example: classifying numbers (Opens a modal)
• Classify numbers: rational & irrational Get 5 of 7 questions to level up!
• Classify numbers Get 5 of 7 questions to level up!

Approximating irrational numbers

• Approximating square roots (Opens a modal)
• Approximating square roots walk through (Opens a modal)
• Comparing irrational numbers with radicals (Opens a modal)
• Approximating square roots to hundredths (Opens a modal)
• Comparing values with calculator (Opens a modal)
• Approximating square roots Get 3 of 4 questions to level up!
• Comparing irrational numbers Get 5 of 7 questions to level up!
• Comparing irrational numbers with a calculator Get 5 of 7 questions to level up!

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9.1.3: Rational and Real Numbers

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Learning Objectives

• Identify the subset(s) of the real numbers that a given number belongs to.
• Locate points on a number line.
• Compare rational numbers.
• Identify rational and irrational numbers.

Introduction

You’ve worked with fractions and decimals, like 3.8 and \(\ 21 \frac{2}{3}\). These numbers can be found between the integer numbers on a number line. There are other numbers that can be found on a number line, too. When you include all the numbers that can be put on a number line, you have the real number line. Let's dig deeper into the number line to see what those numbers look like and where they fall on the number line.

Rational Numbers

The fraction \(\ \frac{16}{3}\), mixed number \(\ 5 \frac{1}{3}\), and decimal 5.33... (or \(\ 5 . \overline{3}\)) all represent the same number. This number belongs to a set of numbers that mathematicians call rational numbers . Rational numbers are numbers that can be written as a ratio of two integers. Regardless of the form used, \(\ 5 . \overline{3}\) is rational because this number can be written as the ratio of 16 over 3, or \(\ \frac{16}{3}\).

Examples of rational numbers include the following.

0.5, as it can be written as \(\ \frac{1}{2}\)

\(\ 2 \frac{3}{4}\), as it can be written as \(\ \frac{11}{4}\)

\(\ -1.6\), as it can be written as \(\ -1 \frac{6}{10}=\frac{-16}{10}\)

\(\ 4\), as it can be written as \(\ \frac{4}{1}\)

-10, as it can be written as \(\ \frac{-10}{1}\)

All of these numbers can be written as the ratio of two integers.

You can locate these points on the number line.

In the following illustration, points are shown for 0.5 or \(\ \frac{1}{2}\), and for 2.75 or \(\ 2 \frac{3}{4}=\frac{11}{4}\).

As you have seen, rational numbers can be negative. Each positive rational number has an opposite. The opposite of \(\ 5 . \overline{3}\) is \(\ -5 . \overline{3}\), for example.

Be careful when placing negative numbers on a number line. The negative sign means the number is to the left of 0, and the absolute value of the number is the distance from 0. So to place -1.6 on a number line, you would find a point that is |-1.6| or 1.6 units to the left of 0. This is more than 1 unit away, but less than 2.

Place \(\ -\frac{23}{5}\) on a number line.

It's helpful to first write this improper fraction as a mixed number: 23 divided by 5 is 4 with a remainder of 3, so \(\ -\frac{23}{5}\) is \(\ -4 \frac{3}{5}\).

Since the number is negative, you can think of it as moving \(\ 4 \frac{3}{5}\) units to the left of 0. \(\ -4 \frac{3}{5}\) will be between -4 and -5.

Which of the following points represents \(\ -1 \frac{1}{4}\)?

• Incorrect. This point is just over 2 units to the left of 0. The point should be 1.25 units to the left of 0. The correct answer is point B.
• Correct. Negative numbers are to the left of 0, and \(\ -1 \frac{1}{4}\) should be 1.25 units to the left. Point B is the only point that’s more than 1 unit and less than 2 units to the left of 0.
• Incorrect. Notice that this point is between 0 and and the first unit mark to the left of 0, so it represents a number between -1 and 0. The point for \(\ -1 \frac{1}{4}\) should be 1.25 units to the left of 0. You may have correctly found 1 unit to the left, but instead of continuing to the left another 0.25 unit, you moved right. The correct answer is point B.
• Incorrect. Negative numbers are to the left of 0, not to the right. The point for \(\ -1 \frac{1}{4}\) should be 1.25 units to the left of 0. The correct answer is point B.
• Incorrect. This point is 1.25 units to the right of 0, so it has the correct distance but in the wrong direction. Negative numbers are to the left of 0. The correct answer is point B.

Comparing Rational Numbers

When two whole numbers are graphed on a number line, the number to the right on the number line is always greater than the number on the left.

The same is true when comparing two integers or rational numbers. The number to the right on the number line is always greater than the one on the left.

Here are some examples.

Which of the following are true?

Option \(\ \text { 1. }-4.1>3.2\)

Option \(\ \text { 2. }-3.2>-4.1\)

Option \(\ \text { 3. }3 .2>4.1\)

Option \(\ \text { 4. }-4.6<-4.1\)

• Option 1 and Option 4
• Option 1 and Option 2
• Option 2 and Option 3
• Option 2 and Option 4
• Options 1, 2, and 3
• Incorrect. -4.6 is to the left of -4.1, so -4.6<-4.1. However, positive numbers such as 3.2 are always to the right of negative numbers such as -4.1, so 3.2>-4.1 or -4.1<3.2. The correct answer is ii and iv, -3.2>-4.1 and -4.6<-4.1.
• Incorrect. -3.2 is to the right of -4.1, so -3.2>-4.1. However, positive numbers such as 3.2 are always to the right of negative numbers such as -4.1, so 3.2>-4.1 or -4.1<3.2. The correct answer is ii and iv, -3.2>-4.1 and -4.6<-4.1.
• Incorrect. -3.2 is to the right of -4.1, so -3.2>-4.1. However, 3.2 is to the left of 4.1, so 3.2<4.1. The correct answer is ii and iv, -3.2>-4.1 and -4.6<-4.1.
• Correct. -3.2 is to the right of -4.1, so -3.2>-4.1. Also, -4.6 is to the left of -4.1, so -4.6<-4.1.
• Incorrect. -3.2 is to the right of -4.1, so -3.2>-4.1. However, positive numbers such as 3.2 are always to the right of negative numbers such as -4.1, so 3.2>-4.1 or -4.1<3.2. Also, 3.2 is to the left of 4.1, so 3.2<4.1. The correct answer is ii and iv, -3.2>-4.1 and -4.6<-4.1.

Irrational and Real Numbers

There are also numbers that are not rational. Irrational numbers cannot be written as the ratio of two integers.

Any square root of a number that is not a perfect square, for example \(\ \sqrt{2}\), is irrational. Irrational numbers are most commonly written in one of three ways: as a root (such as a square root), using a special symbol (such as \(\ \pi\)), or as a nonrepeating, nonterminating decimal.

Numbers with a decimal part can either be terminating decimals or nonterminating decimals . Terminating means the digits stop eventually (although you can always write zeros at the end). For example, 1.3 is terminating, because there’s a last digit. The decimal form of \(\ \frac{1}{4}\) is 0.25. Terminating decimals are always rational.

Nonterminating decimals have digits (other than 0) that continue forever. For example, consider the decimal form of \(\ \frac{1}{3}\), which is 0.3333 ... The 3s continue indefinitely. Or the decimal form of \(\ \frac{1}{11}\), which is 0.090909 ...: the sequence "09" continues forever.

In addition to being nonterminating, these two numbers are also repeating decimals . Their decimal parts are made of a number or sequence of numbers that repeats again and again. A nonrepeating decimal has digits that never form a repeating pattern. The value of \(\ \sqrt{2}\), for example, is 1.414213562... No matter how far you carry out the numbers, the digits will never repeat a previous sequence.

If a number is terminating or repeating, it must be rational; if it is both nonterminating and nonrepeating, the number is irrational.

Is -82.91 rational or irrational?

The set of real numbers is made by combining the set of rational numbers and the set of irrational numbers. The real numbers include natural numbers or counting numbers , whole numbers, integers, rational numbers (fractions and repeating or terminating decimals), and irrational numbers. The set of real numbers is all the numbers that have a location on the number line.

Sets of Numbers

Natural numbers 1, 2, 3, ...

Whole numbers 0, 1, 2, 3, ...

Integers ..., -3, -2, -1, 0, 1, 2, 3, ...

Rational numbers: numbers that can be written as a ratio of two integers—rational numbers are terminating or repeating when written in decimal form

Irrational numbers: numbers that cannot be written as a ratio of two integers—irrational numbers are nonterminating and nonrepeating when written in decimal form

Real numbers: any number that is rational or irrational

What sets of numbers does 32 belong to?

What sets of numbers does \(\ 382 . \overline{3}\) belong to?

What sets of numbers does \(\ -\sqrt{5}\) belong to?

Which of the following sets does \(\ \frac{-33}{5}\) belong to?

whole numbers

rational numbers

irrational numbers

real numbers

• rational numbers only
• irrational numbers only
• rational and real numbers
• irrational and real numbers
• integers, rational numbers, and real numbers
• whole numbers, integers, rational numbers, and real numbers
• Incorrect. The number is rational (it's written as a ratio of two integers) but it's also real. All rational numbers are also real numbers. The correct answer is rational and real numbers, because all rational numbers are also real.
• Incorrect. Irrational numbers can't be written as a ratio of two integers. The correct answer is rational and real numbers, because all rational numbers are also real.
• Correct. The number is between integers, so it can't be an integer or a whole number. It's written as a ratio of two integers, so it's a rational number and not irrational. All rational numbers are real numbers, so this number is rational and real.
• Incorrect. The number is between integers, not an integer itself. The correct answer is rational and real numbers.
• Incorrect. The number is between integers, so it can't be an integer or a whole number. The correct answer is rational and real numbers.

The set of real numbers is all numbers that can be shown on a number line. This includes natural or counting numbers, whole numbers, and integers. It also includes rational numbers, which are numbers that can be written as a ratio of two integers, and irrational numbers, which cannot be written as a the ratio of two integers. When comparing two numbers, the one with the greater value would appear on the number line to the right of the one with the lesser value.